If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional analysis can be defined as the practice of studying relations between physical quantities by recognising the dimensions of them. These dimensions are independent of numerical multiples and similar constants. The quantities present in the world can be described as a function of the fundamental dimensions.

Dimensional Formula

Dimensional formula of a quantity is the expression indicating the powers wherein the fundamental units are to be increased to obtain one unit of a derived quantity. For instance, if Q is an unit of a derived quantity which is defined by Q = M^aL^bT^c, then M^aL^bT^c is known as the dimensional formula with exponents a, b and, c that are called the dimensions.

Dimensional Formula for Resistance

The dimensional formula of resistance can be expressed as:

M¹L²T^-3I^-2

Here,

M = Mass
I = Current
L = Length
T = Time

Resistance (R) = Voltage × Current^-1… [1]

Now, because, Voltage (V) = Electric Field × Distance = [Force × Charge^-1] × Distance

Accordingly, dimensional formula of charge = current × time = I¹T¹

Dimensional formula of voltage = [Force × Charge^-1] × Distance

= [M¹ L¹ T^-2] × [I¹T¹]^-1 × [L¹]

= [M¹ L² T^-3I^-1] … [2]

On replacing equation (ii) in equation (i) we get,

Resistance (R) = Voltage × Current^-1

R = [M¹ L² T^-3I^-1] × [I]^-1 = [M¹ L² T^-3I^-2]

Hence, resistance is dimensionally depicted as M L² T^-3I^-2.

Dimensional Formula of Capacitance

The dimensional formula of Capacitance is expressed by:

M^-1L^-2T⁴I²

Here,

M = Mass
I = Current
L = Length
T = Time

Derivation of Capacitance

Capacitance (C) = Charge × Voltage^-1. . . [1]

Since, Charge = Current × Time

∴ Therefore, the dimensional formula of charge can be give as = [I¹T¹] . . . . [2]

And, Voltage = Electric Field × Distance . . [3]

Now, as per Electric Field,

Electric Field = [Force × Charge^-1]

Hence, dimensional formula of force and charge equals [M¹ L¹ T^-2] and [I¹T¹] respectively.

∴ The dimensional formula of Electric Field = [M¹ L¹ T^-2] × [I¹T¹]^-1

= [M¹ L¹ T^-3 I^-1] . . . [4]

On replacing equation [4] in equation [3] we can obtain,

The dimensional formula of Voltage = [M¹ L¹ T^-3 I^-1] × [L¹]

= [M¹ L² T^-3I^-1] . . . [5]

On replacing equations [5] and [2] in equation [1] we can obtain,

Hence, Capacitance = Charge × Voltage^-1

Or, as can be expressed, C = [I¹T¹] × [M¹ L² T^-3I^-1]^-1 = [M^-1 L^-2 T⁴I²]

Thus, the Capacitance can be dimensionally expressed as [M^-1 L^-2 T⁴I²].

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If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

CBSE CLASS XII Related Questions

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The number of free electrons per unit volume ($\text{m}^3$) in a wire made of copper (resistivity $1.72 \times 10^{-8}\,\Omega\,\text{m}$) is $8.5 \times 10^{28}$. Calculate the average number of collisions made by an electron in one second when an electric field is maintained in the wire.

3.
A magnet held vertically, with its north pole down, is dropped along the axis of a closed solenoid placed vertically on a table. If the observer looks down from the top,

4.
An electron is revolving around a proton in a circular orbit of radius $r$. Using classical physics, prove that the ratio of value of the angular momentum and the value of magnetic moment of electron revolving in the orbit is a constant.

5.
Draw the plot of potential energy of a pair of nucleons as a function of their separation. Write two important conclusions that can be drawn from this plot.

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Consider the nuclear reaction \( X \to Y + Z \). Let \( M_x \), \( M_y \), and \( M_z \) be the masses of the three nuclei X, Y, and Z respectively. Then which of the following relations hold true?

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If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

CBSE CLASS XII Related Questions

2.The number of free electrons per unit volume ($\text{m}^3$) in a wire made of copper (resistivity $1.72 \times 10^{-8}\,\Omega\,\text{m}$) is $8.5 \times 10^{28}$. Calculate the average number of collisions made by an electron in one second when an electric field is maintained in the wire.

3.A magnet held vertically, with its north pole down, is dropped along the axis of a closed solenoid placed vertically on a table. If the observer looks down from the top,

4.An electron is revolving around a proton in a circular orbit of radius $r$. Using classical physics, prove that the ratio of value of the angular momentum and the value of magnetic moment of electron revolving in the orbit is a constant.

5.Draw the plot of potential energy of a pair of nucleons as a function of their separation. Write two important conclusions that can be drawn from this plot.

6.Consider the nuclear reaction \( X \to Y + Z \). Let \( M_x \), \( M_y \), and \( M_z \) be the masses of the three nuclei X, Y, and Z respectively. Then which of the following relations hold true?

Similar Physics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

2.
The number of free electrons per unit volume ($\text{m}^3$) in a wire made of copper (resistivity $1.72 \times 10^{-8}\,\Omega\,\text{m}$) is $8.5 \times 10^{28}$. Calculate the average number of collisions made by an electron in one second when an electric field is maintained in the wire.

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A magnet held vertically, with its north pole down, is dropped along the axis of a closed solenoid placed vertically on a table. If the observer looks down from the top,

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An electron is revolving around a proton in a circular orbit of radius $r$. Using classical physics, prove that the ratio of value of the angular momentum and the value of magnetic moment of electron revolving in the orbit is a constant.

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Draw the plot of potential energy of a pair of nucleons as a function of their separation. Write two important conclusions that can be drawn from this plot.

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Consider the nuclear reaction \( X \to Y + Z \). Let \( M_x \), \( M_y \), and \( M_z \) be the masses of the three nuclei X, Y, and Z respectively. Then which of the following relations hold true?